Variational analysis for options with stochastic volatility and multiple factors

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and multiple factors

Joseph Frédéric Bonnans, Axel Kröner

To cite this version:

Joseph Frédéric Bonnans, Axel Kröner. Variational analysis for options with stochastic volatility

and multiple factors. SIAM Journal on Financial Mathematics, Society for Industrial and Applied

Mathematics 2018, 9 (2), pp.465-492. �hal-01516011v3�

VOLATILITY AND MULTIPLE FACTORS

2

J. FR ´ED ´ERIC BONNANS^† AND AXEL KR ¨ONER^‡ 3

Abstract. This paper performs a variational analysis for a class of European or American 4

options with stochastic volatility models, including those of Heston and Achdou-Tchou. Taking into 5

account partial correlations and the presence of multiple factors, we obtain the well-posedness of the 6

related partial differential equations, in some weighted Sobolev spaces. This involves a generalization 7

of the commutator analysis introduced by Achdou and Tchou in [3].

8

Key words. finance, options, partial differential equations, variational formulation, parabolic 9

variational inequalities 10

AMS subject classifications. 35K20, 35K85, 91G80 11

1. Introduction. In this paper we consider variational analysis for the par-

12

tial differential equations associated with the pricing of European or American op-

13

tions. For an introduction to these models, see Fouque et al., [11]. We will set up

14

a general framework of variable volatility models, which is in particular applicable

15

on the following standard models which are well established in mathematical finance.

16

The well-posedness of PDE formulations of variable volatility poblems was studied in

17

[2, 3, 1, 18], and in the recent work [9, 10].

18

Let the W

i

(t) be Brownian motions on a filtered probability space. The variable

19

s denotes a financial asset, and the components of y are factors that influence the

20

volatility:

21

(i) The Achdou-Tchou model [3], see also Achdou, Franchi, and Tchou [1]:

22

(1.1)

( ds(t) = rs(t)dt + σ(y(t))s(t)dW

1

(t), dy(t) = θ(µ − y(t))dt + ν dW

₂

(t),

23

with the interest rate r, the volatility coefficient σ function of the factor y

24

whose dynamics involves a parameter ν > 0, and positive constants θ and µ.

25

(ii) The Heston model [14]

26

(1.2)

( ds(t) = s(t)

rdt + p

y(t)dW

₁

(t) , dy(t) = θ(µ − y(t))dt + ν p

y(t)dW

2

(t).

27

(iii) The Double Heston model, see Christoffersen, Heston and Jacobs [17], and

28

also Gauthier and Possama¨ı [12]:

29

(1.3)



 

 

ds(t) = s(t)

rdt + p

y

1

(t)dW

1

(t) + p

y

2

(t)dW

2

(t) , dy

₁

(t) = θ

₁

(µ

₁

− y

₁

(t))dt + ν

₁

p

y

₁

(t)dW

₃

(t), dy

2

(t) = θ

2

(µ

2

− y

2

(t))dt + ν

2

p y

2

(t)dW

4

(t).

30

∗Submitted to the editors DATE.

Funding:The first author was suported by the Laboratoire de Finance des March´es de l’Energie, Paris, France. Both authors were supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, in a joint call with Gaspard Monge Pro- gram for optimization, operations research and their interactions with data sciences.

†CMAP, Ecole Polytechnique, CNRS, Universit´e Paris Saclay, and Inria, France (Fred- eric.Bonnans@inria.fr).

‡Department of Mathematics, Humboldt University of Berlin, Berlin, Germany; CMAP, Ecole Polytechnique, CNRS, Universit´e Paris Saclay, and Inria, France (axel.kroener@math.hu-berlin.de)

1

In the last two models we have similar interpretations of the coefficients;

31

in the double Heston model, denoting by h·, ·i the correlation coefficients, we

32

assume that there are correlations only between W

1

and W

3

, and W

2

and W

4

.

33

Consider now the general multiple factor model

34

(1.4) ds = rs(t)dt + P

N

k=1

f

k

(y

k

(t))s

^βk

(t)dW

k

(t),

dy

_k

= θ

_k

(µ

_k

− y

_k

(t))dt + g

_k

(y

_k

(t))dW

_N+k

(t), k = 1, . . . , N.

35

Here the y

_k

are volatility factors, f

_k

(y

_k

) represents the volatility coefficient due to

36

y

_k

, g

_k

(y

_k

) is a volatility coefficient in the dynamics of the kth factor with positive

37

constants θ

_k

and µ

_k

. Let us denote the correlation between the ith and j th Brownian

38

motions by κ

_ij

: this is a measurable function of (s, y, t) with value in [0, 1] (here

39

s ∈ (0, ∞) and y

_k

belongs to either (0, ∞) or R ), see below. We asssume that we have

40

nonzero correlations only between the Brownian motions W

k

and W

N+k

, for k = 1

41

to N , i.e.

42

(1.5) κ

ij

= 0 if i 6= j and |j − i| 6= N.

43

Note that, in some of the main results, we will assume for the sake of simplicity that

44

the correlations are constant.

45

We apply the developed analysis to a subclass of stochastic volatility models, obtained

46

by assuming that κ is constant and

47

(1.6) |f

k

(y

k

)| = |y

k

|

^γk

; |g

k

(y

k

)| = ν

k

|y

k

|

^1−γk

; β

k

∈ (0, 1]; ν

k

> 0; γ

k

∈ (0, ∞).

48

This covers in particular a variant of the Achdou and Tchou model with multiple

49

factors (VAT), when γ

k

= 1, as well as a generalized multiple factor Heston model

50

(GMH), when γ

k

= 1/2, i.e., for k = 1 to N :

51

(1.7) VAT: f

k

(y

k

) = y

k

, g

k

(y

k

) = ν

k

, GMH: f

k

(y

k

) = √

y

_k

, g

k

(y

k

) = ν

k

√ y

_k

.

52

For a general class of stochastic volatility models with correlation we refer to Lions

53

and Musiela [16].

54

The main contribution of this paper is variational analysis for the pricing equa-

55

tion corresponding to the above general class in the sense of the Feynman-Kac theory.

56

This requires in particular to prove continuity and coercivity properties of the corre-

57

sponding bilinear form in weighted Sobolev spaces H and V , respectively, which have

58

the Gelfand property and allow the application of the Lions and Magenes theory [15]

59

recalled in Appendix A and the regularity theory for parabolic variational inequalities

60

recalled in Appendix B. A special emphasis is given to the continuity analysis of the

61

rate term in the pricing equation. Two approaches are presented, the standard one

62

and an extension of the one based on the commutator of first-order differential oper-

63

ators as in Achdou and Tchou [3], extended to the Heston model setting by Achdou

64

and Pironneau [18]. Our main result is that the commutator analysis gives stronger

65

results for the subclass defined by (1.6), generalizing the particular cases of the VAT

66

and GMH classes, see remarks 6.2 and 6.4. In particular we extend some of the results

67

by [3].

68

This paper is organized as follows. In section 2 we give the expression of the bi-

69

linear form associated with the original PDE, and check the hypotheses of continuity

70

and semi-coercivity of this bilinear form. In section 3 we show how to refine this anal-

71

ysis by taking into account the commutators of the first-order differential operators

72

associated with the variational formulation. In section 4 we show how to compute

73

the weighting function involved in the bilinear form. In section 5 we develop the re-

74

sults for a general class introduced in the next section. In section 6 we specialize the

75

results to stochastic volatility models. The appendix recalls the main results of the

76

variational theory for parabolic equations, with a discussion on the characterization

77

of the V functional spaces in the case of one dimensional problems.

78

Notation. We assume that the domain Ω of the PDEs to be considered in the

79

sequel of this paper has the following structure. Let (I, J) be a partition of {0, . . . , N },

80

with 0 ∈ J, and

81

(1.8) Ω :=

^N

Π

k=0

Ω

k

; with Ω

k

:=

R when k ∈ I, (0, ∞) when k ∈ J .

82

Let L

⁰

(Ω) denote the space of measurable functions over Ω. For a given weighting

83

function ρ : Ω → R of class C

¹

, with positive values, we define the weighted space

84

(1.9) L

^2,ρ

(Ω) := {v ∈ L

⁰

(Ω);

Z

Ω

v(x)

²

ρ(x)dx < ∞},

85

which is a Hilbert space endowed with the norm

86

(1.10) kvk

_ρ

:=

Z

Ω

v(x)

²

ρ(x)dx

1/2 87

.

By D(Ω) we denote the space of C

^∞

functions with compact support in Ω. By

88

H

_loc²

(Ω) we denote the space of functions over Ω whose product with an element of

89

D(Ω) belongs to the Sobolev space H

²

(Ω).

90

Besides, let Φ be a vector field over Ω (i.e., a mapping Ω → R

ⁿ

). The first-order

91

differential operator associated with Φ is, for u : Ω → R the function over Ω defined

92

by

93

(1.11) Φ[u](x) :=

n

X

i=0

Φ

i

(x) ∂u

∂x

_i

(x), for all x ∈ Ω.

94

2. General setting. Here we give compute the bilinear form associated with

95

the original PDE, in the setting of the general multiple factor model (1.4). Then we

96

will check the hypotheses of continuity and semi-coercivity of this bilinear form.

97

2.1. Variational formulation. We compute the bilinear form of the variational

98

setting, taking into account a general weight function. We wil see how to choose the

99

functional spaces for a given ρ, and then how to choose the weight itself.

100

2.1.1. The elliptic operator. In financial models the underlying is solution of

101

stochastic differential equations of the form

102

dX(t) = b(t, X (t))dt +

nσ

X

i=1

σ

_i

(t, X (t))dW

_i

. (2.1)

103 104

Here X(t) takes values in Ω, defined in (1.8). That is, X

1

corresponds to the s variable,

105

and X

k+1

, for k = 1 to N, corresponds to y

k

. We have that n

σ

= 2N .

106

So, b and σ

i

, for i = 1 to n

σ

, are mappings (0, T ) × Ω → R

ⁿ

, and the W

i

, for

107

i = 1 to n

σ

, are standard Brownian processes with correlation κ

ij

: (0, T ) × Ω → R

108

between W

i

and W

j

for i, j ∈ {1, . . . , n

σ

}. The n

σ

× n

σ

symmetric correlation matrix

109

κ(·, ·) is nonnegative with unit diagonal:

110

(2.2) κ(t, x) 0; κ

ii

= 1, i = 1, . . . , n

σ

, for a.a. (t, x) ∈ (0, T ) × Ω.

111

Here, for symmetric matrices B and C of same size, by ”C B” we mean that C −B

112

is positive semidefinite. The expression of the second order differential operator A

113

corresponding to the dynamics (2.1) is, skipping the time and space arguments, for

114

u: (0, T ) × Ω → R :

115

(2.3) Au := ru − b · ∇u −

¹₂

n_σ

X

i,j=1

κ

ij

σ

^>_j

u

xx

σ

i

,

116

where

117

(2.4) σ

_j^>

u

xx

σ

i

:=

nσ

X

k,`=1

σ

kj

∂u

²

∂x

k

∂x

`

σ

`i

,

118

r(x, t) represents an interest rate, and u

_xx

is the matrix of second derivatives in space

119

of u. The associated backward PDE for a European option is of the form

120

(2.5)

( − u(t, x) + ˙ A(t, x)u(t, x) = f (t, x), (t, x) ∈ (0, T ) × Ω;

u(x, T ) = u

T

(x), x ∈ Ω,

121

with ˙ u the notation for the time derivative of u, u

T

(x) payoff at final time (horizon)

122

T and the r.h.s. f (t, x) represents dividends (often equal to zero).

123

In case of an American option we obtain a variational inequality; for details we

124

refer to Appendix D.

125

2.1.2. The bilinear form. In the sequel we assume that

126

(2.6) b, σ, κ are C

¹

mappings over [0, T ] × Ω.

127

Multiplying (2.3) by the test function v ∈ D(Ω) and the continuously differentiable

128

weight function ρ: Ω → R and integrating over the domain we can integrate by parts;

129

since v ∈ D(Ω) there will be no contribution from the boundary. We obtain

130

(2.7) −

¹₂

Z

Ω

σ

^>_j

u

_xx

σ

_i

vκ

_ij

ρ =

3

X

p=0

a

^p_ij

(u, v),

131

with

132

(2.8) a

⁰_ij

(u, v) :=

¹₂

Z

Ω n

X

k,`=1

σ

kj

σ

`i

∂u

∂x

k

∂v

∂x

`

κ

ij

ρ =

¹₂

Z

Ω

σ

j

[u]σ

i

[v]κ

ij

ρ,

133 134

(2.9) a

¹_ij

(u, v) :=

¹₂

Z

Ω n

X

k,`=1

σ

kj

σ

`i

∂u

∂x

k

∂(κ

ij

ρ)

∂x

`

v =

¹₂

Z

Ω

σ

j

[u]σ

i

[κ

ij

ρ] v ρ ρ,

135 136

(2.10) a

²_ij

(u, v) :=

¹₂

Z

Ω n

X

k,`=1

σ

kj

∂(σ

`i

)

∂x

`

∂u

∂x

k

vκ

ij

ρ =

¹₂

Z

Ω

σ

j

[u](div σ

i

)vκ

ij

ρ,

137

138

(2.11) a

³_ij

(u, v) :=

¹₂

Z

Ω n

X

k,`=1

∂(σ

kj

)

∂x

_`

σ

`i

∂u

∂x

_k

vκ

ij

ρ =

¹₂

Z

Ω n

X

k=1

σ

i

[σ

kj

] ∂u

∂x

_k

vκ

ij

ρ.

139

Also, for the contributions of the first and zero order terms resp. we get

140

(2.12) a

⁴

(u, v) := − Z

Ω

b[u]vρ; a

⁵

(u, v) :=

Z

Ω

ruvρ.

141

Set

142

(2.13) a

^p

:=

n_σ

X

i,j=1

a

^p_ij

, p = 0, . . . , 3.

143

The bilinear form associated with the above PDE is

144

(2.14) a(u, v) :=

5

X

p=0

a

^p

(u, v).

145

From the previous discussion we deduce that

146

Lemma 2.1. Let u ∈ H

_`oc²

(Ω) and v ∈ D(Ω). Then we have that

147

(2.15) a(u, v) =

Z

Ω

A(t, x)u(x)v(x)ρ(x)dx.

148

2.1.3. The Gelfand triple. We can view a

⁰

as the principal term of the bilinear

149

form a(u, v). Let σ denote the n × n

_σ

matrix whose σ

_j

are the columns. Then

150

(2.16) a

⁰

(u, v) =

n_σ

X

i,j=1

Z

Ω

σ

j

[u]σ

i

[v]κ

ij

ρ = Z

Ω

∇u

^>

σκσ

^>

∇vρ.

151

Since κ 0, the above integrand is nonnegative when u = v; therefore, a

⁰

(u, u) ≥ 0.

152

When κ is the identity we have that a

⁰

(u, u) is equal to the seminorm a

⁰⁰

(u, u), where

153

(2.17) a

⁰⁰

(u, u) :=

Z

Ω

|σ

^>

∇u|

²

ρ.

154

In the presence of correlations it is natural to assume that we have a coercivity of the

155

same order. That is, we assume that

156

(2.18) For some γ ∈ (0, 1]: σκσ

^>

γσσ

^>

, for all (t, x) ∈ (0, T ) × Ω.

157

Therefore, we have

158

(2.19) a

⁰

(u, u) ≥ γa

⁰⁰

(u, u).

159

Remark 2.2. Condition (2.18) holds in particular if

160

(2.20) κ γI,

161

but may also hold in other situations, e.g., when n = 1, n

σ

= 2, κ

12

= 1, and

162

σ

1

= σ

2

= 1. Yet when the σ

i

are linearly independent, (2.19) is equivalent to (2.20).

163

We need to choose a pair (V, H) of Hilbert spaces satisfying the Gelfand condi-

164

tions for the variational setting of Appendix A, namely V densely and continuously

165

embedded in H , a(·, ·) continuous and semi-coercive over V . Additionally, the r.h.s.

166

and final condition of (2.5) should belong to L

²

(0, T ; V

^∗

) and H resp. (and for the

167

second parabolic estimate, to L

²

(0, T ; H ) and V resp. ).

168

We do as follows: for some measurable function h : Ω → R

+

to be specified later

169

we define

170

(2.21)







H := {v ∈ L

⁰

(Ω); hv ∈ L

^2,ρ

(Ω)},

V := {v ∈ H ; σ

i

[v] ∈ L

^2,ρ

(Ω), i = 1, . . . , n

σ

}, V := {closure of D(Ω) in V},

171

endowed with the natural norms,

172

(2.22) kvk

H

:= khvk

ρ

; kuk

²_V

:= a

⁰⁰

(u, u) + kuk

²_H

.

173

We do not try to characterize the space V since this is problem dependent.

174

Obviously, a

⁰

(u, v) is a bilinear continuous form over V. We next need to choose

175

h so that a(u, v) is a bilinear and semi-coercive continuous form, and u

T

∈ H .

176

2.2. Continuity and semi-coercivity of the bilinear form over V. We will

177

see that the analysis of a

⁰

to a

²

is relatively easy. It is less obvious to analyze the

178

term

179

(2.23) a

³⁴

(u, v) := a

³

(u, v) + a

⁴

(u, v).

180

Let ¯ q

_ij

(t, x) ∈ R

ⁿ

be the vector with kth component equal to

181

(2.24) q ¯

ijk

:= κ

ij

σ

i

[σ

kj

].

182

Set

183

(2.25) q ˆ :=

n_σ

X

i,j=1

¯

q

ij

, q := ˆ q − b.

184

Then by (2.11)-(2.12), we have that

185

(2.26) a

³⁴

(u, v) =

Z

Ω

q[u]vρ.

186

We next need to assume that it is possible to choose η

_k

in L

⁰

((0, T ) × Ω), for k = 1

187

to n

_σ

, such that

188

(2.27) q =

nσ

X

k=1

η

_k

σ

_k

.

189

Often the n × n

_σ

matrix σ(t, x) has a.e. rank n. Then the above decomposition is

190

possible. However, the choice for η is not necessarily unique. We will see in examples

191

how to do it. Consider the following hypotheses:

192

h

_σ

≤ c

_σ

h, where h

_σ

:=

nσ

X

i,j=1

|σ

_i

[κ

_ij

ρ]/ρ + κ

_ij

div σ

_i

| , a.e., for some c

_σ

> 0, (2.28)

193

h

_r

≤ c

_r

h, where h

_r

:= |r|

^1/2

, a.e., for some c

_r

> 0, (2.29)

194

h

η

≤ c

η

h, where h

η

:= |η|, a.e., for some c

η

> 0.

(2.30)

195196

Remark 2.3. Let us set for any differentiable vector field Z : Ω → R

ⁿ

197

(2.31) G

ρ

(Z) := div Z + Z[ρ]

ρ .

198

Since κ

_ii

= 1, (2.28) implies that

199

(2.32) |G

ρ

(σ

i

)| ≤ c

σ

h, i = 1; . . . , n

σ

.

200

Remark 2.4. Since

201

(2.33) σ

_i

[κ

_ij

ρ] = σ

_i

[κ

_ij

]ρ + σ

_i

[ρ]κ

_ij

,

202

and |κ

_ij

| ≤ 1 a.e., a sufficient condition for (2.28) is that there exist a positive con-

203

stants c

⁰_σ

such that

204

(2.34) h

⁰_σ

≤ c

⁰_σ

h; h

⁰_σ

:=

n_σ

X

i,j=1

|σ

i

[κ

ij

]| +

n_σ

X

i=1

(|div σ

i

| + |σ

i

[ρ]/ρ|) .

205

We will see in section 4 how to choose the weight ρ so that |σ

i

[ρ]/ρ| can be easily

206

estimated as a function of σ.

207

Lemma 2.5. Let (2.28)-(2.30) hold. Then the bilinear form a(u, v) is both (i)

208

continuous over V , and (ii) semi-coercive, in the sense of (A.5).

209

Proof. (i) We have that a

¹

+ a

²

is continuous, since by (2.9)-(2.10), (2.28) and

210

the Cauchy-Schwarz inequality:

211

(2.35)

|a

¹

(u, v) + a

²

(u, v)| ≤

nσ

X

i,j=1

|a

¹_ij

(u, v) + a

²_ij

(u, v)|

≤

nσ

X

j=1

kσ

j

[u]k

ρ nσ

X

i=1

k(σ

i

[κ

_ij

ρ]/ρ + κ

_ij

div σ

_i

) vk

_ρ

≤ c

σ

n

σ

kvk

H nσ

X

j=1

kσ

j

[u]k

ρ

.

212

213

(ii) Also, a

³⁴

is continuous, since by (2.27) and (2.30):

214

(2.36) |a

³⁴

(u, v)| ≤

n_σ

X

k=1

kσ

k

[u]k

ρ

kη

k

vk

ρ

≤ c

η

kvk

H n_σ

X

k=1

kσ

k

[u]k

ρ

.

215

Set c := c

σ

n

σ

+ c

²_η

. By (2.35)–(2.36), we have that

216

(2.37)

( |a

⁵

(u, v)| ≤ k|r|

^1/2

uk

2,ρ

k|r|

^1/2

vk

2,ρ

≤ c

²_r

kuk

H

kvk

H

,

|a

¹

(u, v) + a

²

(u, v) + a

³⁴

(u, v)| ≤ ca

⁰⁰

(u)

^1/2

kvk

H

.

217

Since a

⁰

is obviously continuous, the continuity of a(u, v) follows.

218

(iii) Semi-coercivity. Using (2.37) and Young’s inequality, we get that

219

(2.38)

a(u, u) ≥ a

0

(u, u) −

a

¹

(u, u) + a

²

(u, u) + a

³⁴

(u, u) −

a

⁵

(u, u)

≥ γa

⁰⁰

(u) − ca

⁰⁰

(u)

^1/2

kuk

_H

− c

_r

kuk

²_H

≥

¹₂

γa

⁰⁰

(u) −

1 2

c

²

γ + c

r

kuk

²_H

,

220

which means that a is semi-coercive.

221

The above consideration allow to derive well-posedness results for parabolic equations

222

and parabolic variational inequalities.

223

Theorem 2.6. (i) Let (V, H) be given by (2.21), with h satisfying (2.28)-(2.30),

224

(f, u

T

) ∈ L

²

(0, T ; V

^∗

)×H. Then equation (2.5) has a unique solution u in L

²

(0, T ; V )

225

with u ˙ ∈ L

²

(0, T ; V

^∗

), and the mapping (f, u

T

) 7→ u is nondecreasing. (ii) If in

226

addition the semi-symmetry condition (A.8) holds, then u in L

^∞

(0, T ; V ) and u ˙ ∈

227

L

²

(0, T ; H ).

228

Proof. This is a direct consequence of Propositions A.1, A.2 and C.1.

229

We next consider the case of parabolic variational inequalities associated with the set

230

(2.39) K := {ψ ∈ V : ψ(x) ≥ Ψ(x) a.e. in Ω},

231

where Ψ ∈ V . The strong and weak formulations of the parabolic variational in-

232

equality are defined in (B.2) and (B.5) resp. The abstract notion of monotonicity is

233

discussed in appendix B. We denote by K the closure of K in V .

234

Theorem 2.7. (i) Let the assumptions of theorem 2.6 hold, with u

T

∈ K. Then

235

the weak formulation (B.5) has a unique solution u in L

²

(0, T ; K) ∩ C(0, T ; H ), and

236

the mapping (f, u

T

) 7→ u is nondecreasing.

237

(ii) Let in addition the semi-symmetry condition (A.8) be satisfied. Then u is the

238

unique solution of the strong formulation (B.2), belongs to L

^∞

(0, T ; V ), and u ˙ belongs

239

to L

²

(0, T ; H ).

240

Proof. This follows from Propositions B.1 and C.2.

241

3. Variational analysis using the commutator analysis. In the following a

242

commutator for first order differential operators is introduced, and calculus rules are

243

derived.

244

3.1. Commutators. Let u : Ω → R be of class C

²

. Let Φ and Ψ be two vector

245

fields over Ω, both of class C

¹

. Recalling (1.11), we may define the commutator of

246

the first-order differential operators associated with Φ and Ψ as

247

(3.1) [Φ, Ψ][u] := Φ[Ψ[u]] − Ψ[Φ[u]].

248

Note that

249

(3.2) Φ[Ψ[u]] =

n

X

i=1

Φ

i

∂(Ψu)

∂x

i

=

n

X

i=1

Φ

i n

X

k=1

∂Ψ

k

∂x

i

∂u

∂x

k

+ Ψ

k

∂

²

u

∂x

k

∂x

i

! .

250

So, the expression of the commutator is

251

(3.3)

[Φ, Ψ] [u] =

n

X

i=1

Φ

i n

X

k=1

∂Ψ

k

∂x

_i

∂u

∂x

_k

− Ψ

i n

X

k=1

∂Φ

k

∂x

_i

∂u

∂x

_k

!

=

n

X

k=1 n

X

i=1

Φ

i

∂Ψ

_k

∂x

i

− Ψ

i

∂Φ

_k

∂x

i

! ∂u

∂x

k

.

252

It is another first-order differential operator associated with a vector field (which

253

happens to be the Lie bracket of Φ and Ψ, see e.g.[4]).

254

3.2. Adjoint. Remembering that H was defined in (2.21), given two vector fields

255

Φ and Ψ over Ω, we define the spaces

256

V(Φ, Ψ) := {v ∈ H ; Φ[v], Ψ[v] ∈ H } , (3.4)

257

V (Φ, Ψ) := {closure of D(Ω) in V (Φ, Ψ)} . (3.5)

258259

We define the adjoint Φ

^>

of Φ (view as an operator over say C

^∞

(Ω, R ), the latter

260

being endowed with the scalar product of L

^2,ρ

(Ω)), by

261

(3.6) hΦ

^>

[u], vi

_ρ

= hu, Φ[v]i

_ρ

for all u, v ∈ D(Ω),

262

where h·, ·i

_ρ

denotes the scalar product in L

^2,ρ

(Ω). Thus, there holds the identity

263

(3.7) Z

Ω

Φ

^>

[u](x)v(x)ρ(x)dx = Z

Ω

u(x)Φ[v](x)ρ(x)dx for all u, v ∈ D(Ω).

264

Furthermore,

265

(3.8)

Z

Ω

u

n

X

i=1

Φ

_i

∂v

∂x

i

ρdx = −

n

X

i=1

Z

Ω

v ∂

∂x

i

(uρΦ

_i

)dx

= −

n

X

i=1

Z

Ω

v ∂

∂x

_i

(uΦ

i

) + u ρ Φ

i

∂ρ

∂x

_i

ρdx.

266

Hence,

267

(3.9) Φ

^>

[u] = −

n

X

i=1

∂

∂x

i

(uΦ

_i

) − uΦ

_i

∂ρ

∂x

i

/ρ = −u div Φ − Φ[u] − uΦ[ρ]/ρ.

268

Remembering the definition of G

_ρ

(Φ) in (2.31), we obtain that

269

(3.10) Φ[u] + Φ

^>

[u] + G

_ρ

(Φ)u = 0.

270

3.3. Continuity of the bilinear form associated with the commutator.

271

Setting, for v and w in V (Φ, Ψ):

272

(3.11) ∆(u, v) :=

Z

Ω

[Φ, Ψ][u](x)v(x)ρ(x)dx,

273

we have

274

(3.12)

∆(u, v) = Z

Ω

(Φ[Ψ[u]]v − Ψ[Φ[u]]v)ρdx = Z

Ω

Ψ[u]Φ

^>

[v] − Φ[u]Ψ

^>

[v])ρdx

= Z

Ω

(Φ[u]Ψ[v] − Ψ[u]Φ[v]) ρdx + Z

Ω

(Φ[u]G

ρ

(Ψ)v − Ψ[u]G

ρ

(Φ)v) ρdx.

275

Lemma 3.1. For ∆(·, ·) to be a continuous bilinear form on V (Φ, Ψ), it suffices

276

that, for some c

_∆

> 0:

277

(3.13) |G

ρ

(Φ)| + |G

ρ

(Ψ)| ≤ c

_∆

h a.e.,

278

and we have then:

279

(3.14) |∆(u, v)| ≤ kΨ[u]k

_ρ

kΦ[v]k

_ρ

+ c

∆

kvk

_H

+ kΦ[u]k

_ρ

kΨ[v]k

_ρ

+ c

∆

kvk

_H

.

280

Proof. Apply the Cauchy Schwarz inequality to (3.12), and use (3.13) combined

281

with the definition of the space H .

282

We apply the previous results with Φ := σ

i

, Ψ := σ

j

. Set for v, w in V :

283

(3.15) ∆

ij

(u, v) :=

Z

Ω

[σ

i

, σ

j

][u](x)v(x)ρ(x)dx, i, j = 1, . . . , n

σ

.

284

We recall that V was defined in (2.21).

285

Corollary 3.2. Let (2.28) hold. Then the ∆

ij

(u, v), i, j = 1, . . . , n

σ

, are con-

286

tinuous bilinear forms over V .

287

Proof. Use remark 2.3 and conclude with lemma 3.1.

288

3.4. Redefining the space H. In section 2.2 we have obtained the continuity

289

and semi-coercivity of a by decomposing q, defined in (2.26), as a linear combination

290

(2.27) of the σ

_i

. We now take advantage of the previous computation of commutators

291

and assume that, more generally, instead of (2.27), we can decompose q in the form

292

(3.16) q =

n_σ

X

k=1

η

_k⁰⁰

σ

_k

+ X

1≤i<j≤nσ

η

⁰_ij

[σ

_i

, σ

_j

] a.e.

293

We assume that η

⁰

and η

⁰⁰

are measurable functions over [0, T ] × Ω, that η

⁰

is weakly

294

differentiable, and that for some c

⁰_η

> 0:

295

(3.17) h

⁰_η

≤ c

⁰_η

h, where h

⁰_η

:= |η

⁰⁰

| +

N

X

i,j=1

σ

_i

[η

⁰_ij

]

a.e., η

⁰

∈ L

^∞

(Ω).

296

Lemma 3.3. Let (2.28), (2.29), and (3.17) hold. Then the bilinear form a(u, v)

297

defined in (2.14) is both (i) continuous and (ii) semi-coercive over V .

298

Proof. (i) We only have to analyze the contribution of a

³⁴

(defined in (2.23)),

299

since the other contributions to a(·, ·) do not change. For the terms in the first sum

300

in (3.16) we have, as was done in (2.36):

301

(3.18)

Z

Ω

σ

_k

[u]η

⁰⁰_k

vρ

≤ kσ

_k

[u]k

_ρ

kσ

_k

[u]η

_k⁰⁰

vk

_ρ

≤ kσ

_k

[u]k

_ρ

kvk

_H

.

302

(ii) Setting w := η

_ij⁰

v and taking here (Φ, Ψ) = (σ

i

, σ

j

), we get that

303

(3.19)

Z

Ω

η

⁰_ij

[σ

i

, σ

j

)[u]vρ = ∆(u, w),

304

where ∆(·, ·) was defined in (3.11). Combining with lemma 3.1, we obtain

305

(3.20)

|∆

ij

(u, v)| ≤ kσ

j

[u]k

_ρ

kσ

i

[w]k

_ρ

+ c

σ

kη

_ij⁰

k

_∞

kvk

_H

+ kσ

i

[u]k

_ρ

kσ

j

[w]k

_ρ

+ c

σ

kη

_ij⁰

k

_∞

kvk

_H

.

306

Since

307

(3.21) σ

i

[η

_ij⁰

v] = η

_ij⁰

σ

i

[v] + σ

i

[η

⁰_ij

]v,

308

by (3.17):

309

(3.22) kσ

i

[w]k

_ρ

≤ kη

_ij⁰

k

_∞

kσ

i

[v]k

_ρ

+

σ

i

[η

_ij⁰

]v

_ρ

≤ kη

⁰_ij

k

_∞

kσ

i

[v]k

_ρ

+ c

η

kvk

H

.

310

Combining these inequalities, point (i) follows.

311

(ii) Use u = v in (3.21) and (3.12). We find after cancellation in (3.12) that

312

(3.23)

∆

ij

(u, η

⁰_ij

u) = Z

Ω

u(σ

i

[u]σ

j

[η

_ij⁰

] − σ

j

[u]σ

i

(η

⁰_ij

))ρ +

Z

Ω

(σ

i

[u]G

ρ

(σ

j

) − σ

j

[u]G

ρ

(σ

i

)) η

_ij⁰

uρ.

313

By (3.17), an upper bound for the absolute value of the first integral is

314

(3.24)

kσ

_i

[u]k

_ρ

+ kσ

_j

[u]k

_ρ

khuk

_ρ

≤ 2 kuk

_V

kuk

_H

.

315

With (2.28), we get an upper bound for the absolute value of the second integral in

316

the same way, so, for any ε > 0:

317

(3.25) |∆

ij

(u, η

⁰_ij

u)| ≤ 4 kuk

_V

kuk

_H

.

318

We finally have that for some c > 0

319

(3.26)

a(u, u) ≥ a

0

(u, u) − c kuk

_V

kuk

_H

,

≥ a

0

(u, u) −

¹₂

kuk

²_V

−

¹₂

c

²

kuk

²_H

,

=

¹₂

kuk

²_V

−

¹₂

(c

²

+ 1) kuk

²_H

.

320

The conclusion follows.

321

Remark 3.4. The statements analogous to theorems 2.6 and 2.7 hold, assuming

322

now that h satisfies (2.28), (2.29), and (3.17) (instead of (2.28)-(2.30)).

323

4. The weight ρ. Classes of weighting functions characterized by their growth

324

are introduced. A major result is the independence of the growth order of the function

325

h on the choice of the weighting function ρ in the class under consideration.

326

4.1. Classes of functions with given growth. In financial models we usually

327

have nonnegative variables and the related functions have polynomial growth. Yet,

328

after a logarithmic transformation, we get real variables whose related functions have

329

exponential growth. This motivates the following definitions.

330

We remind that (I, J ) is a partition of {0, . . . , N }, with 0 ∈ J and that Ω was

331

defined in (1.8).

332

Definition 4.1. Let γ

⁰

and γ

⁰⁰

belong to R

^N+⁺¹

, with index from 0 to N. Let

333

G(γ

⁰

, γ

⁰⁰

) be the class of functions ϕ : Ω → R such that for some c > 0:

334

(4.1) |ϕ(x)| ≤ c

k∈I

Π (e

^γk0xk

+ e

^−γk00xk

) Π

k∈J

(x

^γ

0 k

k

+ x

^−γ

00 k

k

)

.

335

We define G as the union of G(γ

⁰

, γ

⁰⁰

) for all nonnegative (γ

⁰

, γ

⁰⁰

). We call γ

_k⁰

and γ

_k⁰⁰

336

the growth order of ϕ, w.r.t. x

_k

, at −∞ and +∞ (resp. at zero and +∞).

337

Observe that the class G is stable by the operations of sum and product, and that

338

if f , g belong to that class, so does h = f g, h having growth orders equal to the sum

339

of the growth orders of f and g. For a ∈ R , we define

340

(4.2) a

⁺

:= max(0, a); a

⁻

:= max(0, −a); N(a) := (a

²

+ 1)

^1/2

,

341

as well as

342

(4.3) ρ := ρ

I

ρ

J

,

343

where

344

ρ

I

(x ) := Π

k∈I

e

^{−α0kN(x+k)−α00kN(x−k)}

, (4.4)

345

ρ

J

(x ) := Π

k∈J

x

^α

0 k

k

1 + x

^α_k^0k+α00k

, (4.5)

346 347

for some nonnegative constants α

⁰_k

, α

⁰⁰_k

, to be specified later.

348

Lemma 4.2. Let ϕ ∈ G(γ

⁰

, γ

⁰⁰

). Then ϕ ∈ L

^1,ρ

(Ω) whenever ρ is as above, with α

349

satisfying, for some positive ε

⁰

and ε

⁰⁰

, for all k = 0 to N :

350

(4.6)

( α

⁰_k

= ε

⁰

+ γ

_k⁰

, α

⁰⁰_k

= ε

⁰⁰

+ γ

_k⁰⁰

, k ∈ I, α

⁰_k

= (ε

⁰

+ γ

_k⁰⁰

− 1)

₊

, α

⁰⁰_k

= 1 + ε

⁰⁰

+ γ

_k⁰

, k ∈ J.

351

In addition we can choose for k = 0 (if element of J ):

352

(4.7)

( α

⁰₀

:= (ε

⁰

+ γ

₀⁰⁰

− 1)

+

; α

⁰⁰₀

:= 0 if ϕ(s, y) = 0 when s is far from 0, α

⁰₀

:= 0, α

⁰⁰₀

:= 1 + ε

⁰⁰

+ γ

₀⁰

, if ϕ(s, y) = 0 when s is close to 0.

353

Proof. It is enough to prove (4.6), the proof of (4.7) is similar. We know that ϕ

354

satisfy (4.1) for some c > 0 and γ. We need to check the finiteness of

355

(4.8)

Z

Ω

Π

k∈I

(e

^γk0yk

+ e

^−γk00yk

) Π

k∈J

(y

^γ

0 k

k

+ y

^−γ

00 k

k

)

ρ(s, y)d(s, y).

356

But the above integral is equal to the product p

I

p

J

with

357

p

_I

:= Π

k∈I

Z

R

(e

^γk0xk

+ e

^−γ00kxk

)e

^{−α0kN(x+k)−α00kN(x−k)}

dx

_k

, (4.9)

358

p

J

:= Π

k∈J

Z

R+

x

^α

0 k+γ⁰_k

k

+ x

^α

0 k−γ⁰⁰_k k

1 + x

^α

0 k+α⁰⁰_k k

dx

k

. (4.10)

359 360

Using (4.6) we deduce that p

I

is finite since for instance

361

(4.11)

Z

R+

(e

^γ0kxk

+ e

^−γk00xk

)e

^{−α0kN(x+k)−α00kN(x−k)}

dx

k

≤ 2 Z

R+

e

^γk0xk

e

^{−(1+γk0)xk}

dx

k

= 2 Z

R+

e

^−xk

dx

k

= 2,

362

and p

J

is finite since

363

(4.12) p

J

= Π

k∈J

Z

R+

x

^ε

0+γ_k⁰+γ_k⁰⁰

k

+ x

^ε_k⁰⁻¹

1 + x

^ε0+ε00+γ

0 k+γ⁰⁰_k k

dx

k

< ∞.

364

The conclusion follows.

365

4.2. On the growth order of h. Set for all k

366

(4.13) α

_k

:= α

⁰_k

+ α

⁰⁰_k

.

367

Remember that we take ρ in the form (4.3)-(4.4).

368

Lemma 4.3. We have that:

369

(i) We have that

370

(4.14)

ρ

x_k

ρ

∞

≤ α

k

, k ∈ I;

x ρ ρ

x_k

∞

≤ α

k

, k ∈ J.

371

(ii) Let h satisfying either (2.28)-(2.30) or (2.28)-(2.29), and (3.17). Then the growth

372

order of h does not depend on the choice of the weighting function ρ.

373

Proof. (i) For k ∈ I this is an easy consequence of the fact that N (·) is non

374

expansive. For k ∈ J , we have that

375

(4.15) x

ρ ρ

_xk

= x ρ

α

⁰_k

x

^α0k−1

(1 + x

^αk

) − x

^α0k

α

_k

x

^αk−1

(1 + x

^αk

)

²

= α

⁰_k

− α

⁰⁰_k

x

^αk

1 + x

^αk

.

376

We easily conclude, discussing the sign of the numerator.

377

(ii) The dependence of h w.r.t. ρ is only through the last term in (2.28), namely,

378

P

i

|σ

_i

[ρ]/ρ. By (i) we have that

379

(4.16)

σ

_i^k

[ρ]

ρ

≤

ρ

x_k

ρ

∞

|σ

^k_i

| ≤ α

k

|σ

_i^k

|, k ∈ I,

380 381

(4.17)

σ

^k_i

[ρ]

ρ

≤

x

_k

ρ

_xk

ρ

_∞

σ

_i^k

x

k

≤ α

k

σ

^k_i

x

k

, k ∈ J.

382

In both cases, the choice of α has no influence on the growth order of h.

383

4.3. European option. In the case of a European option with payoff u

T

(x),

384

we need to check that u

T

∈ H , that is, ρ must satisfy

385

(4.18)

Z

Ω

|u

T

(x)|

²

h(x)

²

ρ(x)dx < ∞.

386

In the framework of the semi-symmetry hypothesis (A.8), we need to check that

387

u

T

∈ V , which gives the additional condition

388

(4.19)

nσ

X

i=1

Z

Ω

|σ

i

[u

_T

](x)|

²

ρ(x)dx < ∞.

389

In practice the payoff depends only on s and this allows to simplify the analysis.

390

5. Applications using the commutator analysis. The commutator analysis

391

is applied to the general multiple factor model and estimates for the function h char-

392

acterizing the space H (defined in (2.21)) are derived. The estimates are compared

393

to the case when the commutator analysis is not applied. The resulting improvement

394

wil be established in the next section.

395

5.1. Commutator and continuity analysis. We analyze the general multiple

396

factor model (1.4), which belongs to the class of models (2.1) with Ω ⊂ R

^1+N

, n

σ

=

397

2N , and for i = 1 to N :

398

(5.1) σ

i

[v] = f

i

(y

i

)s

^βi

v

s

; σ

N+i

[v] = g

i

(y

i

)v

i

,

399

with f

_i

and g

_i

of class C

¹

over Ω. We need to compute the commutators of the first-

400

order differential operators associated with the σ

_i

. The correlations will be denoted

401

by

402

(5.2) κ ˆ

k

:= κ

k,N+k

, k = 1, . . . , N.

403

Remark 5.1. We use many times the following rule. For Ω ⊂ R

ⁿ

, where n =

404

1 + N, u ∈ H

¹

(Ω), a, b ∈ L

⁰

, and vector fields Z[u] := au

x₁

and Z

⁰

[u] := bu

x₂

, we

405

have Z [Z

⁰

[u]] = a(bu

x₂

)

x₁

= ab

x₁

u

x₂

+ abu

x₁x₂

, so that

406

(5.3) [Z, Z

⁰

][u] = ab

_x1

u

_x2

− ba

_x2

u

_x1

.

407

We obtain that

408

(5.4) [σ

i

, σ

`

][u] = (β

`

− β

i

)f

i

(y

i

)f

`

(y

`

)s

^βi+β`−1

u

s

, 1 ≤ i < ` ≤ N,

409 410

(5.5) [σ

_i

, σ

_N+i

][u] = −s

^βi

f

_i⁰

(y

_i

)g

_i

(y

_i

)u

_s

, i = 1, . . . , N,

411

and

412

(5.6) [σ

i

, σ

N+`

][u] = [σ

N+i

, σ

N+`

][u] = 0, i 6= `.

413

Also,

414

(5.7)

div σ

i

+ σ

i

[ρ]

ρ = f

i

(y

i

)s

^βi−1

(β

i

+ s ρ

s

ρ ), div σ

_N+i

+ σ

_N+i

[ρ]

ρ = g

_i⁰

(y

_i

) + g

_i

(y

_i

) ρ

_i

ρ .

415

5.1.1. Computation of q. Remember the definitions of ¯ q, ˆ q and q in (2.24) and

416

(2.25), where δ

ij

denote the Kronecker operator. We obtain that, for 1 ≤ i, j, k ≤ N:

417

(5.8)



 



 



¯

q

_ij0

= δ

_ij

β

_j

f

_i²

(y

_i

)s

^2βi−1

; q ¯

_iik

= 0;

¯

q

_i,N+j

= 0;

¯

q

_N+i,j,0

= δ

_ij

κ ˆ

_i

f

⁰

(y

_i

)g

_i

(y

_i

)s

^βi

; q ¯

_N+i,j,k

= 0;

¯

q

_N+i,N+j,k

= δ

_ijk

g

_i

(y

_i

)g

_i⁰

(y

_i

).

418

That means, we have for ˆ q = P

2N

i,j=1

q ¯

ij

and q = ˆ q − b that

419

(5.9) ˆ q

0

= P

N

i=1

β

i

f

_i²

(y

i

)s

^2βi−1

+ ˆ κ

i

f

⁰

(y

i

)g

i

(y

i

)s

^βi

; q

0

= ˆ q

0

− rs, ˆ

q

k

= g

k

(y

k

)g

⁰_k

(y

k

); q

k

= ˆ q

k

− θ

k

(µ

k

− y

k

), k = 1, . . . , N.

420

5.2. Computation of η

⁰

and η

⁰⁰

. The coefficients η

⁰

, η

⁰⁰

are solution of (3.16).

421

We can write η = ˆ η + ˜ η, where

422

(5.10)

ˆ q =

n_σ

X

i=1

ˆ

η

⁰⁰_i

σ

i

+ X

1≤i,j≤n_σ

ˆ

η

_ij⁰

[σ

i

, σ

j

], η ˆ

_ij⁰

= 0 if i = j.

−b =

nσ

X

i=1

˜

η

⁰⁰_i

σ

_i

+ X

1≤i,j≤nσ

˜

η

_ij⁰

[σ

_i

, σ

_j

], η ˜

_ij⁰

= 0 if i = j.

423

For k = 1 to N, this reduces to

424

(5.11)

( η ˆ

_N⁰⁰_+k

g

_k

(y

_k

) = g

_k⁰

(y

_k

)g

_k

(y

_k

);

˜

η

_N⁰⁰_+k

g

k

(y

k

) = −θ

k

(µ

k

− y

k

).

425

So, we have that

426

(5.12)





 ˆ

η

⁰⁰_N+k

= g

⁰_k

(y

k

);

˜

η

⁰⁰_N+k

= −θ

k

(µ

_k

− y

_k

) g

k

(y

k

) .

427

For the 0th component, (5.10) can be expressed as

428

(5.13)



 

 

 

 

 

 

 

 

 

 

N

X

k=1

−ˆ η

_k,N+k⁰

f

_k⁰

(y

_k

)g

_k

(y

_k

)s

^βk

− ˆ κ

_k

f

_k⁰

(y

_k

)g

_k

(y

_k

)s

^βk

+

N

X

k=1

ˆ

η

⁰⁰_k

f

k

(y

k

)s

^βk

− β

k

f

_k²

(y

k

)s

^2βk−1

+

N

X

k=1

−˜ η

⁰_k,N+k

f

_k⁰

(y

k

)g

k

(y

k

)s

^βk

+ ˜ η

_k⁰⁰

f

k

(y

k

)s

^βk

− rs = 0.

429

We choose to set each term in parenthesis in the first two lines above to zero. It

430

follows that

431

ˆ

η

_k,N+k⁰

= −ˆ κ

k

∈ L

^∞

(Ω), η ˆ

_k⁰⁰

= β

k

f

k

(y

k

)s

^βk−1

. (5.14)

432433

If N > 1 we (arbitrarily) choose then to set the last line to zero with

434

(5.15) η ˜

⁰⁰_k

= ˜ η

_k⁰

= 0, k = 2, . . . , N.

435

It remains that

436

˜

η

⁰⁰₁

f

1

(y

1

)s

^β1

− η ˜

⁰_1,N+1

f

₁⁰

(y

1

)g

1

(y

1

)s

^β1

= rs.

(5.16)

437438

Here, we can choose to take either ˜ η

₁⁰⁰

= 0 or ˜ η

⁰_1,N+1

= 0. We obtain then two

439

possibilities:

440

(5.17)



 



 



(i) η ˜

₁⁰⁰

= 0 and ˜ η

_1,N+1⁰

= −rs

^1−β1

f

₁⁰

(y

1

)g

1

(y

1

) , (ii) η ˜

₁⁰⁰

= rs

^1−β1

f

1

(y

1

) and ˜ η

⁰_1,N+1

= 0.

441

5.2.1. Estimate of the h function. We decide to choose case (i) in (5.17).

442

The function h needs to satisfy (2.28), (2.29), and (3.17) (instead of (2.30)). Instead

443

of (2.28), we will rather check the stronger condition (2.34). We compute

444

h

⁰_σ

:=

N

X

k=1

|f

k

(y

k

)|s

^βk

|(ˆ κ

k

)

s

| + | ρ

s

ρ |

+ |g

k

(y

k

)|

|(ˆ κ

k

)

k

| + | ρ

k

ρ | (5.18)

445

+

N

X

k=1

β

_k

|f

k

(y

_k

)s

^βk−1

| + |g

_k⁰

(y

_k

)|

,

446

h

r

:= |r|

¹²

, (5.19)

447

h

⁰_η

:= ˆ h

⁰_η

+ ˜ h

⁰_η

, (5.20)

448449

where we have

450

ˆ h

⁰_η

:=

N

X

k=1

β

k

|f

k

(y

k

)|s

^βk−1

+ |g

_k⁰

(y

k

)| +

f

k

(y

k

)|s

^βk

∂ˆ κ

k

∂s

+

g

k

(y

k

) ∂ κ ˆ

k

∂y

k

, (5.21)

451

˜ h

⁰_η

:=

N

X

k=1

θ

k

(µ

k

− y

k

) g

_k

(y

_k

)

+

r f

1

(y

1

) f

₁⁰

(y

₁

)g

₁

(y

₁

)

+

rg

1

(y

1

)s

^1−β1

∂

∂y

₁

1 f

₁⁰

(y

₁

)g

₁

(y

₁

)

. (5.22)

452 453

Remark 5.2. Had we chosen (ii) instead of (i) in (5.17), this would only change

454

the expression of h ˜

⁰_η

that would then be

455

(5.23) ˜ h

⁰_η

=

N

X

k=1

θ

k

(µ

k

− y

k

) g

_k

(y

_k

)

+

rs

^1−β1

f

₁

(y

₁

) .

456

5.2.2. Estimate of the h function without the commutator analysis. The

457

only change in the estimate of h will be the contribution of h

⁰_η

and h

⁰⁰_η

. We have to

458

satisfy (2.28)-(2.30). In addition, ignoring the commutator analysis, we would solve

459

(5.13) with ˆ η

⁰

= 0, meaning that we choose

460

(5.24) η ˆ

⁰⁰_k

:= β

k

f

k

(y

k

)s

^βk−1

+ ˆ κ

k

f

_k⁰

(y

k

)g

k

(y

k

)

f

k

(y

k

) , k = 1, . . . , N,

461

and take ˜ η

⁰⁰₁

out of (5.16). Then condition (3.17), with here ˆ η

⁰

= 0, would give

462

(5.25) h ≥ c

η

h

η

, where h

η

:= h

ηˆ

+ h

η˜

,

463

with

464

h

ηˆ

:=

N

X

k=1

β

k

|f

k

(y

k

)|s

^βk−1

+ |ˆ κ

k

|

f

_k⁰

(y

k

)g

k

(y

k

) f

k

(y

k

)

+ |g

_k⁰

(y

k

)|

, (5.26)

465

h

_η˜

:=

N

X

k=1

θ

_k

(µ

_k

− y

_k

) g

k

(y

k

)

+

rs

^1−β1

f

1

(y

1

) . (5.27)

466 467

We will see in applications that this is in general worse.

468